where \(\mathbf {r=U}^H \mathbf {y}\) , \(\mathbf {\Phi }=\mathbf {U}^H\mathbf {A}=\mathbf {\Lambda }\mathbf {V}\) ,
\(\mathbf {\Lambda }\)  is an \(M\times N\)  rectangular diagonal matrix, and \(\mathbf {\omega } = \mathbf {U}^H \mathbf {w}\)  remains a zero-mean Gaussian noise vector with the same covariance matrix \(\beta ^{-1} \mathbf {I}\) . Applying the vector step size AMP [1]} with model (REF ) leads to the first version of UAMP black(called UAMPv1) shown in Algorithm REF .blackBy replacing \(\mathbf {r}\)  and \(\mathbf {\Phi }\) 
with \(\mathbf {y}\)  and \(\mathbf {A}\)  in Algorithm REF  respectively, the original AMP algorithm is recovered.
black

UAMP (UAMPv2 executes operations in [ ])
Initialize \(\mathbf {\tau }_x^{(0)} (\mathrm {or}~{\tau }_x^{(0)})>0\)  and \({{\mathbf {x}}^{(0)} }\) . Set \(\mathbf {s}^{(-1)}=\mathbf { 0 }\)  and \(t=0\) . Define vector \(\mathbf {\lambda }=\mathbf { \Lambda \Lambda }^H \textbf {1}\) .
